On the Application of Hansbo’s Method for Interface Problems

Abstract

Summary.A geometrically nonlinear finite element framework for the modeling of strong discontinuities in three dimensional continua is presented. By doubling the degrees of freedom in the discontinuous elements, the algorithm allows for arbitrary discontinuities which are not a priori restricted to inter-element boundaries. On both sides of the discontinuity we apply an independent interpolation of the deformation field. Accordingly, the suggested approach relies exclusively on displacement degrees of freedom. On the discontinuity surface itself, we make use of the cohesive zone concept to account for a smooth crack opening. A three dimensional bending problem and the classical symmetric and non-symmetric peel test demonstrate the performance of the suggested method.Key words: FE technology, failure, strong discontinuities, cohesive surfaces MotivationThe modeling of strong discontinuities within the finite element setting has gained increasing attention ever since the pioneering work on extended finite elements by Belytschko and co-workers [1-6], see also Wells et al. [7][8][9]. The extended finite element method is extremely powerful and the underlying concept is remarkably simple: Enrichment functions are applied to interpolate a part of the deformation field which is decomposed in the standard continuous part and an additional discontinuous part. Rather then following the classical X-FEM approach and introducing jumps in the deformation field as additional unknowns, we suggest a slightly modified approach based on the recent ideas by Hansbo and Hansbo [10][11][12]. Their celebrated concept which essentially generalizes the ideas of Nitsche [13] was adopted by Mergheim et al. [14] and has recently been extended to finite deformations in a fully three dimensional setting, see [15].In line with the Hansbo method, a discontinuous approximation is accomplished by doubling the degrees of freedom in the discontinuous elements. The deformation field is then interpolated independently on both sides of Alain Combescure et al. (eds.), IUTAM Symposium on Discretization Methods for Evolving Discontinuities,[255][256][257][258][259][260][261][262][263][264][265]

Modeling three‐dimensional crack propagation—A comparison of crack path tracking strategies

“…[9,12,16]. Figure 15 curves are nearly similar for the four different tracking strategies, the resulting crack surfaces reveal slight differences.…”

Towards the treatment of boundary conditions for global crack path tracking in three-dimensional brittle fracture

“…The underlying split of a tetrahedral element can produce two different combinations of sub-elements depending on whether the crack surface forms a triangle or a quadrilateral, see in detail, e.g., [4,35,42,43,60]. In the former case, we obtain a four-node tetrahedron and a six-node wedge element, whereas in the latter case we obtain two six-node wedge elements.…”

“…2, all particles initially located on the unique discontinuity surface are mapped onto two surfaces γ + and γ − in the deformed configuration. To uniquely characterize discontinuous failure at finite deformations, we apply the concept of a fictitious discontinuity ϕ which is assumed to be located between the two discontinuity surfaces γ + and γ − in the deformed configuration, see, e.g., [4,[41][42][43]51]. Cohesive interface problem: concept of fictitious discontinuity surface located between the two discontinuity surfaces γ + and γ − Again, the corresponding deformation gradient F and its Jacobian J = det F follow straightforwardly.…”

“…On the fictitious discontinuity, we apply the cohesive interface concept, for which all inelastic deformation around the crack tip is collectively represented through the cohesive Cauchy tractions t. Similar to the Cauchy stresses σ in the bulk, the cohesive Cauchy tractions t on the fictitious discontinuity can be related to the cohesive Piola tractions T on the reference domain through Nanson's formula in terms of the area elements da and dA. We conveniently assume a decoupling of the normal and tangential constitutive behavior and introduce the cohesive Cauchy tractions t in the following form, see, e.g., [41][42][43].…”

“…At the additional cost of successively introduced global degrees freedom, smooth discrete cracks could finally be modeled anywhere in the domain, see also [27][28][29][30] for two dimensional computations, as well as [31][32][33][34][35][36] for three dimensional computations. While the XFEM uses the displacement jump as additional global unknown, the method proposed by Hansbo and Hansbo [37][38][39] works exclusively with deformation degrees of freedom, see also [40][41][42][43]. In this manuscript we focus on the latter approach which basically belongs to the category of discontinuous Galerkin methods.…”